{"id":"palomar/PALOMAR-2026-11-03-000031","url":"https://github.com/jkfarland-bot/ConjCoord/tree/c956f4808ba6c6f15a1d4234a04cd8fc56be2711","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-11-03-000031-v1.json","repo":"https://github.com/jkfarland-bot/ConjCoord","tags":["math.NT","msc:11R29","msc:11E25","msc:11F37","msc:11A15","msc:68V20"],"score":0,"title":"ConjCoord: a class number congruence modulo 16 for p = 5 (mod 16) and Conjecture coord of Paper V","trust":"high","domain":"math.NT","source":"palomar","authors":["Joseph S. Farland"],"reviews":0,"subject":"math.NT","summary":"For primes p = 5 (mod 16), p = a^2 + b^2 with b even, the companion note proves h(-8p) = h(-4p) + 8 [b = +-6 (mod 16)] (mod 16), which settles Conjecture 5.3 of Paper V of the K Project series. The proof writes 4(h(-8p) - h(-4p)) as a combination of representation numbers of x^2 + 2y^2 + 4z^2 and x^2 + 2y^2 + 2z^2. It shows that a weight-2 form Phi built from sieved theta series vanishes modulo 256, by Sturm's theorem and a finite check below the bound 512, and reads off the coefficient at p. This development formalizes the proof with the modular input (Serre-Stark modularity and Sturm's…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-11-03","reactions":{},"formalized":true,"formalizes":[{"title":"The first derivative bit at p = 5 (mod 16): a proof of Conjecture coord of Paper V","identifier":"https://doi.org/10.5281/zenodo.23903759","relationship":"formalizes"}]}